Question

using a two - way radio, melissa can talk to anyone within an 8 - mile radius of her location. oscar is 30°west of south of melissas location. what is oscars exact location relative to melissas location, at the greatest distance where they can talk using two - way radios?
a. - 4
b. - 4\sqrt{3}
c. 2\sqrt{3}
d. - 3\sqrt{4}
e. 3
f. - 2
x - coordinate
y - coordinate

Explanation:

Step1: Determine the formula for coordinates

For a point at a distance \(r\) from the origin with an angle \(\theta\) (measured from the positive \(x -\)axis in the counter - clockwise direction), the \(x\) and \(y\) coordinates are given by \(x = r\cos\theta\) and \(y=r\sin\theta\). Here, the angle \(\theta = 240^{\circ}\) (since \(30^{\circ}\) west of south is \(180 + 60=240^{\circ}\) from the positive \(x -\)axis) and \(r = 8\) miles.

Step2: Calculate the \(x\) - coordinate

We know that \(\cos(240^{\circ})=\cos(180^{\circ}+ 60^{\circ})=-\cos(60^{\circ})=-\frac{1}{2}\). Using the formula \(x = r\cos\theta\), with \(r = 8\) and \(\theta = 240^{\circ}\), we have \(x=8\cos(240^{\circ})\). Substituting \(\cos(240^{\circ})=-\frac{1}{2}\), we get \(x = 8\times(-\frac{1}{2})=- 4\).

Step3: Calculate the \(y\) - coordinate

We know that \(\sin(240^{\circ})=\sin(180^{\circ}+60^{\circ})=-\sin(60^{\circ})=-\frac{\sqrt{3}}{2}\). Using the formula \(y = r\sin\theta\), with \(r = 8\) and \(\theta = 240^{\circ}\), we have \(y = 8\sin(240^{\circ})\). Substituting \(\sin(240^{\circ})=-\frac{\sqrt{3}}{2}\), we get \(y=8\times(-\frac{\sqrt{3}}{2})=-4\sqrt{3}\).

Answer:

\(x - coordinate=-4\) (a), \(y - coordinate=-4\sqrt{3}\) (b)